1. **Problem statement:** (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ can be rewritten as $$3 \sin^2 \theta + 8 \sin \theta - 3 = 0$$. (b) Solve for $$0 \leq x \leq 90^\circ$$ the equation $$8 \tan 2x = 3 \cos 2x$$, giving answers to 2 decimal places. 2. **Part (a) - Rewriting the equation:** Start with the given equation: $$8 \tan \theta = 3 \cos \theta$$ Recall that $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$, so substitute: $$8 \frac{\sin \theta}{\cos \theta} = 3 \cos \theta$$ Multiply both sides by $$\cos \theta$$ to clear the denominator: $$8 \sin \theta = 3 \cos^2 \theta$$ 3. Use the Pythagorean identity $$\cos^2 \theta = 1 - \sin^2 \theta$$: $$8 \sin \theta = 3 (1 - \sin^2 \theta)$$ Expand the right side: $$8 \sin \theta = 3 - 3 \sin^2 \theta$$ Bring all terms to one side: $$3 \sin^2 \theta + 8 \sin \theta - 3 = 0$$ This matches the required form. 4. **Part (b) - Solving the equation:** Given: $$8 \tan 2x = 3 \cos 2x$$ Rewrite $$\tan 2x$$ as $$\frac{\sin 2x}{\cos 2x}$$: $$8 \frac{\sin 2x}{\cos 2x} = 3 \cos 2x$$ Multiply both sides by $$\cos 2x$$: $$8 \sin 2x = 3 \cos^2 2x$$ Use $$\cos^2 2x = 1 - \sin^2 2x$$: $$8 \sin 2x = 3 (1 - \sin^2 2x)$$ Expand: $$8 \sin 2x = 3 - 3 \sin^2 2x$$ Bring all terms to one side: $$3 \sin^2 2x + 8 \sin 2x - 3 = 0$$ 5. Let $$y = \sin 2x$$, then solve the quadratic: $$3 y^2 + 8 y - 3 = 0$$ Use the quadratic formula: $$y = \frac{-8 \pm \sqrt{8^2 - 4 \times 3 \times (-3)}}{2 \times 3} = \frac{-8 \pm \sqrt{64 + 36}}{6} = \frac{-8 \pm \sqrt{100}}{6}$$ Calculate: $$y_1 = \frac{-8 + 10}{6} = \frac{2}{6} = \frac{1}{3}$$ $$y_2 = \frac{-8 - 10}{6} = \frac{-18}{6} = -3$$ Since $$\sin 2x$$ must be between -1 and 1, discard $$y_2 = -3$$. 6. Solve for $$x$$: $$\sin 2x = \frac{1}{3}$$ Take inverse sine: $$2x = \sin^{-1} \left( \frac{1}{3} \right)$$ Calculate: $$2x \approx 19.47^\circ$$ Divide by 2: $$x \approx 9.74^\circ$$ 7. Since $$\sin 2x$$ is positive, the general solutions for $$2x$$ in $$0^\circ \leq 2x \leq 180^\circ$$ are: $$2x = 19.47^\circ$$ or $$2x = 180^\circ - 19.47^\circ = 160.53^\circ$$ Divide by 2: $$x \approx 9.74^\circ$$ or $$x \approx 80.27^\circ$$ **Final answers:** (a) $$3 \sin^2 \theta + 8 \sin \theta - 3 = 0$$ (b) $$x \approx 9.74^\circ, 80.27^\circ$$